Do Now Find the area of the given

Slopes and Planar Areas of Polar Curves Section 10. 6 a

Finding the Slope of a Polar Curve The slope of is given by .

Finding the Slope of a Polar Curve provided at The curve has a .

Finding the Slope of a Polar Curve Practice Problem: Find the slope of the

Finding the Slope of a Polar Curve Practice Problem: Find the lines tangent to

Area in the Polar Plane The new challenge: How to find the area of

Area in the Polar Plane Area of region OTS is approximately: Letting get infinitely

Area in the Polar Plane What about the area between two polar curves? :

Guided Practice Find the area of the region in the plane enclosed by the

Guided Practice Find the area of the region that lies inside the circle and

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Do Now Find the area of the given sector:

Slopes and Planar Areas of Polar Curves Section 10. 6 a

Finding the Slope of a Polar Curve The slope of is given by . Observe… First, conceptualize the curve parametrically: The derivative (Chain Rule!): Product Rule!

Finding the Slope of a Polar Curve provided at The curve has a . 1. horizontal tangent at a point where and , 2. vertical tangent at a point where and , (Note: If both derivatives are zero, no conclusion can be drawn without further investigation)

Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point.

Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point. Undefined Check these results graphically!

Finding the Slope of a Polar Curve Practice Problem: Find the slope of the given curve at the indicated point. Check these results graphically!

Finding the Slope of a Polar Curve Practice Problem: Find the lines tangent to the given rose curve at the pole. Start with a graph… is zero at Tangent Lines:

Area in the Polar Plane The new challenge: How to find the area of a polar region TOS… Break it into smaller sectors! Area of a typical sector:

Area in the Polar Plane The new challenge: How to find the area of a polar region TOS… Break it into smaller sectors! Area of region OTS is approximately:

Area in the Polar Plane Area of region OTS is approximately: Letting get infinitely small (summing the areas of infinitely many sectors…) The area of the region between the origin and the curve , , is

Area in the Polar Plane What about the area between two polar curves? : The area of region , , is

Guided Practice Find the area of the region in the plane enclosed by the cardioid: First, check out the graph…

Guided Practice Find the area of the region in the plane enclosed by the cardioid: Confirm this answer numerically!

Guided Practice Find the area of the region that lies inside the circle and outside the cardioid. First, check out the graph…

Guided Practice Find the area of the region that lies inside the circle and outside the cardioid.